\contentsline {chapter}{\tocchapter {Chapter}{}{Introduction}}{7}{section*.1}
\contentsline {section}{\tocsection {}{}{What is this book about}}{7}{section*.3}
\contentsline {section}{\tocsection {}{}{What's in the book}}{7}{section*.4}
\contentsline {section}{\tocsection {}{}{What is syntax and what is semantics?}}{8}{section*.5}
\contentsline {section}{\tocsection {}{}{For whom is the book written}}{8}{section*.6}
\contentsline {section}{\tocsection {}{}{Exercises}}{9}{section*.7}
\contentsline {section}{\tocsection {}{}{How to find a definition in the book}}{9}{section*.8}
\contentsline {chapter}{\tocchapter {Chapter}{1}{Main definitions and basic facts}}{11}{chapter.1}
\contentsline {section}{\tocsection {}{1.1}{Sets}}{11}{section.1.1}
\contentsline {section}{\tocsection {}{1.2}{Words}}{12}{section.1.2}
\contentsline {subsection}{\tocsubsection {}{1.2.1}{The origin of words}}{12}{subsection.1.2.1}
\contentsline {subsection}{\tocsubsection {}{1.2.2}{The free semigroup}}{13}{subsection.1.2.2}
\contentsline {subsection}{\tocsubsection {}{1.2.3}{Orders on words}}{14}{subsection.1.2.3}
\contentsline {subsection}{\tocsubsection {}{1.2.4}{A characterization of free semigroups}}{14}{subsection.1.2.4}
\contentsline {subsection}{\tocsubsection {}{1.2.5}{Commuting words}}{14}{subsection.1.2.5}
\contentsline {section}{\tocsection {}{1.3}{Graphs}}{16}{section.1.3}
\contentsline {subsection}{\tocsubsection {}{1.3.1}{Basic definitions}}{16}{subsection.1.3.1}
\contentsline {subsection}{\tocsubsection {}{1.3.2}{Automata}}{17}{subsection.1.3.2}
\contentsline {subsection}{\tocsubsection {}{1.3.3}{Mealy automata}}{18}{subsection.1.3.3}
\contentsline {subsection}{\tocsubsection {}{1.3.4}{Graphs in the sense of Serre}}{19}{subsection.1.3.4}
\contentsline {subsection}{\tocsubsection {}{1.3.5}{Inverse automata and foldings}}{19}{subsection.1.3.5}
\contentsline {section}{\tocsection {}{1.4}{Universal Algebra}}{20}{section.1.4}
\contentsline {subsection}{\tocsubsection {}{1.4.1}{Basic definitions}}{20}{subsection.1.4.1}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.1.1}{Terms and trees}}{21}{subsubsection.1.4.1.1}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.1.2}{Identities and varieties}}{21}{subsubsection.1.4.1.2}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.1.3}{Examples of universal algebras}}{21}{subsubsection.1.4.1.3}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.1.4}{Congruences in groups, vector spaces and linear algebras}}{25}{subsubsection.1.4.1.4}
\contentsline {subsubsection}{\tocsubsubsection {}{1.4.1.5}{Algebras of transformations}}{25}{subsubsection.1.4.1.5}
\contentsline {subsection}{\tocsubsection {}{1.4.2}{Free algebras in varieties}}{26}{subsection.1.4.2}
\contentsline {subsection}{\tocsubsection {}{1.4.3}{The Birkhoff theorem}}{26}{subsection.1.4.3}
\contentsline {subsection}{\tocsubsection {}{1.4.4}{Locally finite varieties}}{27}{subsection.1.4.4}
\contentsline {subsection}{\tocsubsection {}{1.4.5}{The Burnside problem for varieties of algebras}}{28}{subsection.1.4.5}
\contentsline {subsection}{\tocsubsection {}{1.4.6}{Finitely based and Cross varieties}}{28}{subsection.1.4.6}
\contentsline {subsection}{\tocsubsection {}{1.4.7}{Inherently non-finitely based finite algebras: the link between finite and infinite}}{30}{subsection.1.4.7}
\contentsline {section}{\tocsection {}{1.5}{Growth of algebras}}{33}{section.1.5}
\contentsline {section}{\tocsection {}{1.6}{Symbolic Dynamics}}{34}{section.1.6}
\contentsline {subsection}{\tocsubsection {}{1.6.1}{Basic definitions}}{34}{subsection.1.6.1}
\contentsline {subsection}{\tocsubsection {}{1.6.2}{Subshifts}}{35}{subsection.1.6.2}
\contentsline {section}{\tocsection {}{1.7}{Rewriting systems}}{37}{section.1.7}
\contentsline {subsection}{\tocsubsection {}{1.7.1}{The main definitions}}{37}{subsection.1.7.1}
\contentsline {subsection}{\tocsubsection {}{1.7.2}{Confluence}}{38}{subsection.1.7.2}
\contentsline {subsection}{\tocsubsection {}{1.7.3}{What if a rewriting system is not confluent? The art of Knuth-Bendix}}{41}{subsection.1.7.3}
\contentsline {subsection}{\tocsubsection {}{1.7.4}{String rewriting}}{42}{subsection.1.7.4}
\contentsline {section}{\tocsection {}{1.8}{Presentations of semigroups}}{45}{section.1.8}
\contentsline {subsection}{\tocsubsection {}{1.8.1}{Semigroups and monoids: basic definitions}}{45}{subsection.1.8.1}
\contentsline {subsection}{\tocsubsection {}{1.8.2}{A characterization of free semigroups}}{45}{subsection.1.8.2}
\contentsline {subsection}{\tocsubsection {}{1.8.3}{Free and non-free semigroups}}{46}{subsection.1.8.3}
\contentsline {subsection}{\tocsubsection {}{1.8.4}{Congruences, ideals and quotient semigroups}}{47}{subsection.1.8.4}
\contentsline {subsection}{\tocsubsection {}{1.8.5}{String rewriting and presentations}}{47}{subsection.1.8.5}
\contentsline {subsection}{\tocsubsection {}{1.8.6}{The free group}}{49}{subsection.1.8.6}
\contentsline {subsection}{\tocsubsection {}{1.8.7}{The free group and ping-pong}}{50}{subsection.1.8.7}
\contentsline {subsection}{\tocsubsection {}{1.8.8}{The growth function, growth series and Church-Rosser presentations}}{51}{subsection.1.8.8}
\contentsline {subsection}{\tocsubsection {}{1.8.9}{The Cayley graphs}}{53}{subsection.1.8.9}
\contentsline {chapter}{\tocchapter {Chapter}{2}{Avoidable words}}{55}{chapter.2}
\contentsline {section}{\tocsection {}{2.1}{An old example}}{55}{section.2.1}
\contentsline {section}{\tocsection {}{2.2}{Proof of Thue's theorem}}{56}{section.2.2}
\contentsline {section}{\tocsection {}{2.3}{Square-Free words}}{57}{section.2.3}
\contentsline {section}{\tocsection {}{2.4}{$k$th power-free substitutions}}{58}{section.2.4}
\contentsline {section}{\tocsection {}{2.5}{Avoidable words}}{59}{section.2.5}
\contentsline {subsection}{\tocsubsection {}{2.5.1}{Examples and simple facts}}{59}{subsection.2.5.1}
\contentsline {subsection}{\tocsubsection {}{2.5.2}{Zimin's theorem}}{60}{subsection.2.5.2}
\contentsline {subsection}{\tocsubsection {}{2.5.3}{Fusions, free subsets and free deletions}}{60}{subsection.2.5.3}
\contentsline {subsection}{\tocsubsection {}{2.5.4}{The Bean-Ehrenfeucht-McNulty \& Zimin theorem}}{61}{subsection.2.5.4}
\contentsline {subsection}{\tocsubsection {}{2.5.5}{The Proof of $(3)\rightarrow (2)$}}{61}{subsection.2.5.5}
\contentsline {subsection}{\tocsubsection {}{2.5.6}{The Proof of $(2)\rightarrow (1)$}}{61}{subsection.2.5.6}
\contentsline {subsection}{\tocsubsection {}{2.5.7}{Proof of $(1)\rightarrow (3)$}}{61}{subsection.2.5.7}
\contentsline {subsection}{\tocsubsection {}{2.5.8}{Simultaneous avoidability}}{63}{subsection.2.5.8}
\contentsline {section}{\tocsection {}{2.6}{Further reading}}{63}{section.2.6}
\contentsline {chapter}{\tocchapter {Chapter}{3}{Semigroups}}{65}{chapter.3}
\contentsline {section}{\tocsection {}{3.1}{Structure of semigroups}}{65}{section.3.1}
\contentsline {subsection}{\tocsubsection {}{3.1.1}{Periodic semigroups}}{66}{subsection.3.1.1}
\contentsline {subsection}{\tocsubsection {}{3.1.2}{Periodic semigroups with exactly one idempotent}}{66}{subsection.3.1.2}
\contentsline {subsection}{\tocsubsection {}{3.1.3}{Finite nil-semigroups.}}{67}{subsection.3.1.3}
\contentsline {section}{\tocsection {}{3.2}{Free semigroups and varieties}}{68}{section.3.2}
\contentsline {subsection}{\tocsubsection {}{3.2.1}{Free Rees factor-semigroups}}{68}{subsection.3.2.1}
\contentsline {subsection}{\tocsubsection {}{3.2.2}{A description of relatively free semigroups}}{69}{subsection.3.2.2}
\contentsline {subsubsection}{\tocsubsubsection {}{3.2.2.1}{Examples of varieties of semigroups.}}{69}{subsubsection.3.2.2.1}
\contentsline {section}{\tocsection {}{3.3}{The Burnside problem for varieties}}{70}{section.3.3}
\contentsline {subsection}{\tocsubsection {}{3.3.1}{Subshifts and semigroups}}{71}{subsection.3.3.1}
\contentsline {subsection}{\tocsubsection {}{3.3.2}{An application of subshifts to semigroups}}{73}{subsection.3.3.2}
\contentsline {subsection}{\tocsubsection {}{3.3.3}{Implication $2\rightarrow 4$}}{75}{subsection.3.3.3}
\contentsline {section}{\tocsection {}{3.4}{Brown's theorem and uniformly recurrent\ words}}{80}{section.3.4}
\contentsline {section}{\tocsection {}{3.5}{Burnside problems and the finite basis property}}{81}{section.3.5}
\contentsline {section}{\tocsection {}{3.6}{Inherently non-finitely based finite semigroups}}{81}{section.3.6}
\contentsline {subsection}{\tocsubsection {}{3.6.1}{Some advanced semigroup theory}}{81}{subsection.3.6.1}
\contentsline {subsubsection}{\tocsubsubsection {}{3.6.1.1}{Ideals and $0$-simple semigroups}}{81}{subsubsection.3.6.1.1}
\contentsline {subsubsection}{\tocsubsubsection {}{3.6.1.2}{Semigroups without divisors isomorphic to $A_2$ and $B_2$}}{85}{subsubsection.3.6.1.2}
\contentsline {subsection}{\tocsubsection {}{3.6.2}{The description of inherently non-finitely based\ finite semigroups}}{87}{subsection.3.6.2}
\contentsline {section}{\tocsection {}{3.7}{Growth functions of semigroups}}{94}{section.3.7}
\contentsline {subsection}{\tocsubsection {}{3.7.1}{The definition}}{94}{subsection.3.7.1}
\contentsline {subsection}{\tocsubsection {}{3.7.2}{Chebyshev, Hardy-Ramanujan and semigroups of intermediate growth}}{95}{subsection.3.7.2}
\contentsline {subsubsection}{\tocsubsubsection {}{3.7.2.1}{A semigroup of matrices}}{95}{subsubsection.3.7.2.1}
\contentsline {subsubsection}{\tocsubsubsection {}{3.7.2.2}{A semigroup of automatic transformations}}{97}{subsubsection.3.7.2.2}
\contentsline {subsubsection}{\tocsubsubsection {}{3.7.2.3}{Relatively free semigroups, Zimin words and growth}}{98}{subsubsection.3.7.2.3}
\contentsline {section}{\tocsection {}{3.8}{Inverse semigroups}}{98}{section.3.8}
\contentsline {subsection}{\tocsubsection {}{3.8.1}{Basic facts about inverse semigroups}}{98}{subsection.3.8.1}
\contentsline {subsection}{\tocsubsection {}{3.8.2}{The inverse semigroups of bi-rooted inverse automata}}{99}{subsection.3.8.2}
\contentsline {subsection}{\tocsubsection {}{3.8.3}{Identities of finite inverse semigroups, Zimin words, and symbolic dynamics}}{100}{subsection.3.8.3}
\contentsline {section}{\tocsection {}{3.9}{Synchronizing automata and road coloring}}{103}{section.3.9}
\contentsline {subsection}{\tocsubsection {}{3.9.1}{Complete automata}}{103}{subsection.3.9.1}
\contentsline {subsection}{\tocsubsection {}{3.9.2}{Synchronizing automata}}{103}{subsection.3.9.2}
\contentsline {subsection}{\tocsubsection {}{3.9.3}{The road coloring problem}}{104}{subsection.3.9.3}
\contentsline {subsection}{\tocsubsection {}{3.9.4}{The road coloring theorem to a classification of subshifts}}{113}{subsection.3.9.4}
\contentsline {section}{\tocsection {}{3.10}{Further reading}}{115}{section.3.10}
\contentsline {subsection}{\tocsubsection {}{3.10.1}{Growth}}{115}{subsection.3.10.1}
\contentsline {subsection}{\tocsubsection {}{3.10.2}{The finite basis problem for semigroups}}{115}{subsection.3.10.2}
\contentsline {subsection}{\tocsubsection {}{3.10.3}{The restricted Burnside problem}}{115}{subsection.3.10.3}
\contentsline {subsection}{\tocsubsection {}{3.10.4}{Free semigroups in periodic varieties}}{115}{subsection.3.10.4}
\contentsline {subsection}{\tocsubsection {}{3.10.5}{Inherently non-finitely based semigroups}}{115}{subsection.3.10.5}
\contentsline {subsection}{\tocsubsection {}{3.10.6}{The road coloring conjecture}}{115}{subsection.3.10.6}
\contentsline {subsubsection}{\tocsubsubsection {}{3.10.6.1}{The \v {C}ern\'{y} conjecture}}{116}{subsubsection.3.10.6.1}
\contentsline {chapter}{\tocchapter {Chapter}{4}{Rings}}{119}{chapter.4}
\contentsline {section}{\tocsection {}{4.1}{The basic notions}}{119}{section.4.1}
\contentsline {section}{\tocsection {}{4.2}{Free associative algebras}}{120}{section.4.2}
\contentsline {section}{\tocsection {}{4.3}{Commuting elements in free associative algebras}}{120}{section.4.3}
\contentsline {section}{\tocsection {}{4.4}{Burnside-type problems in associative algebras}}{122}{section.4.4}
\contentsline {subsection}{\tocsubsection {}{4.4.1}{Preliminaries}}{122}{subsection.4.4.1}
\contentsline {subsection}{\tocsubsection {}{4.4.2}{Shirshov's height theorem}}{124}{subsection.4.4.2}
\contentsline {subsection}{\tocsubsection {}{4.4.3}{The Dubnov-Ivanov-Nagata-Higman theorem}}{127}{subsection.4.4.3}
\contentsline {subsection}{\tocsubsection {}{4.4.4}{Golod counterexamples to the Kurosh problem}}{128}{subsection.4.4.4}
\contentsline {subsection}{\tocsubsection {}{4.4.5}{Zimin words and the Baer radical}}{132}{subsection.4.4.5}
\contentsline {section}{\tocsection {}{4.5}{The finite basis problem}}{134}{section.4.5}
\contentsline {subsection}{\tocsubsection {}{4.5.1}{Basic facts about finite associative rings}}{134}{subsection.4.5.1}
\contentsline {subsection}{\tocsubsection {}{4.5.2}{Positive result. Identities of finite rings}}{136}{subsection.4.5.2}
\contentsline {subsection}{\tocsubsection {}{4.5.3}{Negative result}}{140}{subsection.4.5.3}
\contentsline {section}{\tocsection {}{4.6}{Further reading}}{143}{section.4.6}
\contentsline {chapter}{\tocchapter {Chapter}{5}{Groups}}{145}{chapter.5}
\contentsline {section}{\tocsection {}{5.1}{Van Kampen diagrams}}{145}{section.5.1}
\contentsline {subsection}{\tocsubsection {}{5.1.1}{Group presentations}}{145}{subsection.5.1.1}
\contentsline {subsection}{\tocsubsection {}{5.1.2}{Van Kampen diagrams: the definition}}{145}{subsection.5.1.2}
\contentsline {subsection}{\tocsubsection {}{5.1.3}{Van Kampen diagrams and tilings. An elementary school problem and its non-elementary solution}}{148}{subsection.5.1.3}
\contentsline {subsection}{\tocsubsection {}{5.1.4}{The three main methods of dealing with van Kampen diagrams: bands, Swiss cheese, and small cancelation}}{149}{subsection.5.1.4}
\contentsline {subsubsection}{\tocsubsubsection {}{5.1.4.1}{The band method. HNN extensions}}{149}{subsubsection.5.1.4.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.1.4.2}{The Swiss cheese method. Amalgamated products}}{152}{subsubsection.5.1.4.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.1.4.3}{The auxiliary planar graphs. The Dehn-Greendlinger algorithm}}{153}{subsubsection.5.1.4.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.1.4.4}{Small cancellation and conjugacy. Annular (Schupp) diagrams.}}{157}{subsubsection.5.1.4.4}
\contentsline {subsubsection}{\tocsubsubsection {}{5.1.4.5}{Diagrams and elementary topology}}{158}{subsubsection.5.1.4.5}
\contentsline {subsubsection}{\tocsubsubsection {}{5.1.4.6}{Small cancellation, $0$-cells and the baby version of contiguity subdiagrams\ }}{159}{subsubsection.5.1.4.6}
\contentsline {section}{\tocsection {}{5.2}{The Burnside problems for groups}}{161}{section.5.2}
\contentsline {subsection}{\tocsubsection {}{5.2.1}{Golod's counterexample to the unbounded Burnside problem for groups}}{161}{subsection.5.2.1}
\contentsline {subsection}{\tocsubsection {}{5.2.2}{The Bounded Burnside problem. Positive results}}{162}{subsection.5.2.2}
\contentsline {subsection}{\tocsubsection {}{5.2.3}{The Novikov-Adian theorem.}}{164}{subsection.5.2.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.1}{The basic rough idea}}{164}{subsubsection.5.2.3.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.2}{$j$-pairs}}{165}{subsubsection.5.2.3.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.3}{Parameters}}{165}{subsubsection.5.2.3.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.4}{Contiguity subdiagrams. The definition and the spirit of Zimin words}}{165}{subsubsection.5.2.3.4}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.5}{Boundary arcs: smooth, almost geodesic, compatible with a cell}}{167}{subsubsection.5.2.3.5}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.6}{A good system of contiguity subdiagrams\ }}{169}{subsubsection.5.2.3.6}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.7}{The auxiliary weighted graphs and the existence of a cell that sticks out}}{169}{subsubsection.5.2.3.7}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.8}{Why are all diagrams normal?}}{172}{subsubsection.5.2.3.8}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.9}{Why are smooth arcs almost geodesic?}}{173}{subsubsection.5.2.3.9}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.10}{Why do diagrams with small perimeters have small ranks?}}{174}{subsubsection.5.2.3.10}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.11}{Short cuts in annular diagrams}}{174}{subsubsection.5.2.3.11}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.12}{A generalization of the Fine-Wilf theorem}}{174}{subsubsection.5.2.3.12}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.13}{The conclusion of the road map}}{177}{subsubsection.5.2.3.13}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.14}{Why is the group defined by $\@mathcal {PB}$ of exponent $n$}}{177}{subsubsection.5.2.3.14}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.15}{The groups $G_i$ are hyperbolic.}}{178}{subsubsection.5.2.3.15}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3.16}{Why do we need $n$ to be odd?}}{178}{subsubsection.5.2.3.16}
\contentsline {section}{\tocsection {}{5.3}{The finite basis problem for groups}}{178}{section.5.3}
\contentsline {subsection}{\tocsubsection {}{5.3.1}{An example of R. Bryant and Yu. Kleiman}}{178}{subsection.5.3.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.3.1.1}{The result}}{178}{subsubsection.5.3.1.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.3.1.2}{Some properties of nilpotent groups of class 2}}{179}{subsubsection.5.3.1.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.3.1.3}{The group $A_n$}}{179}{subsubsection.5.3.1.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.3.1.4}{The group $B_n$.}}{181}{subsubsection.5.3.1.4}
\contentsline {subsubsection}{\tocsubsubsection {}{5.3.1.5}{The group $C_n$}}{181}{subsubsection.5.3.1.5}
\contentsline {subsection}{\tocsubsection {}{5.3.2}{Construction of the algebra $R$ used in Section\nonbreakingspace \ref {rings without finite identity basis}}}{182}{subsection.5.3.2}
\contentsline {section}{\tocsection {}{5.4}{Groups and identities, Ab\'ert's criterium}}{186}{section.5.4}
\contentsline {section}{\tocsection {}{5.5}{Diagram groups}}{187}{section.5.5}
\contentsline {subsection}{\tocsubsection {}{5.5.1}{The definition of a 2-complex and its fundamental group}}{187}{subsection.5.5.1}
\contentsline {subsection}{\tocsubsection {}{5.5.2}{The Squier complex of a string rewriting system}}{188}{subsection.5.5.2}
\contentsline {subsection}{\tocsubsection {}{5.5.3}{Diagrams as 2-dimensional words, diagram groups}}{188}{subsection.5.5.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.3.1}{Diagram monoids and groups. The definition}}{188}{subsubsection.5.5.3.1}
\contentsline {subsection}{\tocsubsection {}{5.5.4}{Diagram groups and Squier complexes}}{190}{subsection.5.5.4}
\contentsline {subsection}{\tocsubsection {}{5.5.5}{Diagram groups. Examples}}{190}{subsection.5.5.5}
\contentsline {subsection}{\tocsubsection {}{5.5.6}{Combinatorics on diagrams}}{190}{subsection.5.5.6}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.6.1}{Some basic properties of semigroup diagrams}}{191}{subsubsection.5.5.6.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.6.2}{Nice and simple diagrams}}{191}{subsubsection.5.5.6.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.6.3}{Automorphisms of spherical graphs}}{192}{subsubsection.5.5.6.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.6.4}{Commuting diagrams}}{194}{subsubsection.5.5.6.4}
\contentsline {subsection}{\tocsubsection {}{5.5.7}{The R.Thompson group $F$}}{195}{subsection.5.5.7}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.7.1}{The definitions of $F$ }}{195}{subsubsection.5.5.7.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.7.2}{From diagrams to functions}}{196}{subsubsection.5.5.7.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.5.7.3}{From diagrams to pairs of full binary trees.}}{198}{subsubsection.5.5.7.3}
\contentsline {subsection}{\tocsubsection {}{5.5.8}{Multilinear identities of non-associative algebras and elements of $F$}}{199}{subsection.5.5.8}
\contentsline {subsection}{\tocsubsection {}{5.5.9}{$F$ is lawless}}{200}{subsection.5.5.9}
\contentsline {subsection}{\tocsubsection {}{5.5.10}{$F$ does not have non-cyclic free subgroups}}{200}{subsection.5.5.10}
\contentsline {subsection}{\tocsubsection {}{5.5.11}{Two Church-Rosser presentations of $F$.}}{201}{subsection.5.5.11}
\contentsline {section}{\tocsection {}{5.6}{Growth of groups}}{203}{section.5.6}
\contentsline {subsection}{\tocsubsection {}{5.6.1}{Similar groups have similar growth}}{203}{subsection.5.6.1}
\contentsline {subsection}{\tocsubsection {}{5.6.2}{Commutative groups}}{204}{subsection.5.6.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.2.1}{Free commutative groups}}{204}{subsubsection.5.6.2.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.2.2}{Arbitrary finitely generated commutative groups}}{204}{subsubsection.5.6.2.2}
\contentsline {subsection}{\tocsubsection {}{5.6.3}{Nilpotent groups}}{206}{subsection.5.6.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.3.1}{Some basic properties of nilpotent groups}}{206}{subsubsection.5.6.3.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.3.2}{Distorted cyclic subgroups of nilpotent groups}}{207}{subsubsection.5.6.3.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.3.3}{The theorem of Bass and Guivarc'h}}{208}{subsubsection.5.6.3.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.3.4}{The lower bound}}{209}{subsubsection.5.6.3.4}
\contentsline {subsubsection}{\tocsubsubsection {}{5.6.3.5}{The upper bound}}{209}{subsubsection.5.6.3.5}
\contentsline {subsection}{\tocsubsection {}{5.6.4}{Grigorchuk's group of intermediate growth}}{210}{subsection.5.6.4}
\contentsline {section}{\tocsection {}{5.7}{Further reading}}{215}{section.5.7}
\contentsline {subsection}{\tocsubsection {}{5.7.1}{Growth}}{215}{subsection.5.7.1}
\contentsline {subsection}{\tocsubsection {}{5.7.2}{The finite basis problem for varieties of groups}}{215}{subsection.5.7.2}
\contentsline {subsection}{\tocsubsection {}{5.7.3}{The bounded Burnside problem}}{215}{subsection.5.7.3}
\contentsline {subsection}{\tocsubsection {}{5.7.4}{The restricted Burnside problem}}{215}{subsection.5.7.4}
\contentsline {subsection}{\tocsubsection {}{5.7.5}{Zelmanov words and inherently non-finitely based\ varieties of groups}}{215}{subsection.5.7.5}
\contentsline {chapter}{\tocchapter {Chapter}{}{Bibliography}}{217}{theorem.5.7.3}
\contentsline {chapter}{\tocchapter {Chapter}{}{Index}}{225}{chapter*.9}
